Floquet Solution
According to Floquet's theorem (or Bloch's theorem), for fixed values of a,q, Mathieu's equation admits a complex valued solution of form
where is a complex number, the Mathieu exponent, and P is a complex valued function which is periodic in with period . However, P is in general not sinusoidal. In the example plotted below, (real part, red; imaginary part; green):
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